AI vs Numerical PDE - Turbulent Euler/Navier-Stokes - Data Bank

Today the mathematics community has been struck by an OpenAI repository with mathematical proofs of mathematical theorems:

• Several hundred claimed resolutions of long-standing conjectures across number theory, algebraic geometry, representation theory and mathematical physics...
• Milne's rationality conjecture, Goldfeld's conjecture, Nagata's conjecture, Bloch's conjecture for complex surfaces, Fujita's freeness conjecture, Hilbert's tenth problem over ℚ, the irrationality of Catalan's constant. Those are not routine results; several have been open for decades.
• But nothing whatever in numerical analysis, finite elements, error control or computational PDE. That is not a subject being served unevenly. It is a subject that does not appear.

Here is a reflection on this new revolutionary state of affairs: 

• AI as Computational Mathematician Rather Than Theorem Prover

It seems AI as LLM can do analytical mathematics since it is a symbolic language, and then better than human mathematicians, like computer chess vs Karpov. But computational mathematics is a collection of well defined algorithms not asking for AI as LLM, and so can serve mainly for coding as a language, data collection and output evaluation. 

This means that computational mathematics appears as the winner under the AI advance, while the effect  on analytical mathematics may be profound. 

I will now test if AI for Euler/Navier-Stokes can deliver a data bank of representative turbulent flows with dual-based output error control, by running the same code with data collected by AI, and report the result.    

Nobel Prize in Physics 2026

The Nobel Prize in Physics 2026 is another Prize assigned to the neutrino as a "ghost particle" invented by Wolfgang Pauli to describe an apparent loss of energy in beta-decay which nobody could explain. Pauli was not happy with his baby because it had no features at all, no mass, no charge, nothing. This is what the Nobel Prize home page has to say

• The neutrino is the shyest particle in the universe, has no electric charge and almost no mass. In general, it passes unnoticed through matter – seldom does a neutrino make its presence felt by colliding with an atomic nucleus. 
• Every second, without you noticing, 65 billion neutrinos from the Sun flow through your little finger nail.

The 2026 Prize went to detection of a couple of high energy neutrinos from outer space in a one kilometer ice-cube in the South Pole ice mass, not directly because the neutrino is too shy to show but indirectly as a little flash of light supposedly the trace of neutrino flying by. 

Another Parturient montes, nascetur ridiculus mus as theoretical physics, the King of Science, with mathematics the Queen.

This is what RealQM says about neutrinos and beta-decay.

New Explanation of Periodic Table based on Coulomb Alone

RealQM as a new model for atoms based on non-overlapping electron unit charge densities interacting by Coulomb potentials has delivered a new explanation of the Periodic Table based on the numbers 2 and 8 carrying the physics of both period length 2, 8, 8, 18, 18, 32, 32, 50, 50, and period doubling as (8, 8), (18, 18), (32, 32) and (50, 50) in the following form:

• 2,  8, 18=2+8+8, 32 =8+8+8+8, 50 = 2+8+8+8+8+8.

Note that this is a fundamentally different explanation the the textbook one since 100 years based on the s, p, d, e, f orbitals of the eigenfunctions of the Hydrogen atom with numbers 8=2+6, 18=2+6+10, 32=2+6+10+14 without physics. Also recall that the magic/golden  numbers of the nucleus are 2 (alpha-particle) and 8 (O-16) as the most stable nuclei. RealNucleus explains why. 

Read the explanation in this article under revision for IJQC.

Euler's Equations: Symbolic vs Numerical Mathematics

Here is another summary of the situation actualized by the OpenAI 57 page symbolic/analytical proof of blow-up of solutions to the unforced Euler equations:

• Euler Equations: Symbolic vs Numerical Mathematics

The mathematics community is in shock after the OpenAI 167 page proof of blow-up for forced Navier-Stokes equations presenting a solution to the Clay Navier-Stokes Millennium Problem far beyond the horizon of human mathematicians. Same shock as delivered by computer chess 20 years ago.  

What will the impact be on symbolic mathematics as the Queen of Science? What will be left to human mathematicians to do? Something like speed chess? Best proof of given proposition in 5 minutes? And for mathematics education? 

The article pleads for a synthesis of symbolic and numerical mathematics with a role for human mathematicians. What do you think? Game over? Synthesis possible? Math education tomorrow?

Per Enflo 1944 - 2026 Mathematician and Pianist

Per Enflo on September 28 in the middle of the next step on the daily walk with his wife Lena in Östervåla Uppland, under inspection of plants on the ground in the spirit of Linné and with inspiration from the sky finally closing his constructive proof of the Invariant Subspace Problem for Hilbert Spaces, took a last breath sending a shock wave to family, friends and mathematical community. Per Enflo is certainly the most famous Swedish mathematician all times by having solved named major problems, also concert pianist expressing the true meaning of the music of Mozart, Beethoven, Schubert and Chopin.

I met first Per during my post doc years 1974-76 at the math department of the University of Chicago, when Per visited as the new shining star of Functional Analysis, with offers from all the big universities, after having solved one of the key open problems in that area formulated by its founder Stefan Banach in the 1930s, as a 9 page Counterexample to the Approximation Problem in Banach Spaces published in Acta Math in 1973, which took the math community with storm. Watch the documentary movie about Banach with Per in the main role (and me with little side role) and Per's home page.

Per then followed up in 1981 with an 101 page counterexample to the Invariant Subspace Problem in Banach Spaces, published in Acta Math after 6 years of refereeing, to return 40 years later to the case of Hilbert spaces with an explicit construction of an invariant subspace. 

It took 43 years before we met again, in Stockholm in 2008 when Per had rejoined with his love from youth Lena and returned to Sweden after 25 years in the US, and Per welcomed me and my wife Ingrid to his piano trio concert at the Mazer Musical Society. We found each other on the spot into a 20 year long friendship along with our wifes, with music, math and love. It is very sad that Per with his very kind person and amazing talent is no longer here. In the Swedish math community we shared experiences of exclusivity as a special bond.   

Per was a master of constructive mathematics, constructing an invariant subspace for any continuous linear operator T in Hilbert space H, by constructing a sequence of vectors converging to a vector for which repeated application of T does not span H and thus forms an invariant subspace. Per was also a master of constructive music as combination of body/hand and soul/mind with ability to on the spot transpose any given sheet music to any key. We could meet in hands-on constructive math/piano but also lofty speculation, never any argument.

On Sept 8 and 26 another shock wave into the math community had been sent by OpenAI as constructive proofs of blow-up of solutions to the equations of Navier-Stokes (167 pages) and Euler (57 pages) performed by AI. 

Per expressed that he had been lucky to not meet this seemingly formidable competition before, with AI able to construct proofs of any number of pages, beyond understanding by human mathematicians. Per could thus pass on to a heaven of math without shattered beliefs with his usual happy "that is also ok" and with his Musical Legacy completed in 15 CD on Spotify.

Pictures from Aug 20 2026:

 

OpenAI vs Clay Navier-Stokes vs Numerical Analysis

Here are two new articles connecting to the recent hype of AI doing mathematics instead of mathematicians:

• Constructive Solution of the Clay Navier-Stokes Problem
• AI vs Numerical Analysis

Questions to ponder

• Does AI pull the carpet under mathematics education?
• What is the meaning of AI proofs of mathematical theorems?
• Is the main role of a mathematician to prove theorems?
• What is the role of numerical analysis?

No Longer Any Mission for Theoretical Physicists?

Swedish theoretical physics (string theory) Ulf Danielson today in Swedish mainstream media (SvD) concludes that with now AI writing articles and then including whatever theoretical physics is needed, there is no longer any role for theoretical physicists like himself with former prime role to contribute that element to science and education: 

• Research is becoming meaningless.  

Danielson thus gives up completely when confronted with AI, but this surrender has been prepared during 50 years of theoretical physics dominated by string theory, where physics seems to have been replaced by words without meaning. 

But is it true that AI now can take over theoretical physics?  If true, a first task could  then be to find out if string theory makes any sense or not. Asking Claude I do not get a clear answer. This is the same if I ask simple questions about quantum mechanics. It seems that AI is as confused as human theoretical physicists concerning the mysteries of QM, no wonder recalling that AI is trained to propagate the mysteries.

But AI has a capacity as LLM showing to surpass that of humans, if only left alone, namely to follow the logic of language. With that ability it should be possible to sort out the mysteries of QM resulting from violating logic and so give a new start 100 years after QM was born. 

RealQM represents such a new start realized by me as human interacting with AI. My impression is that AI could not have done that by itself. A new idea was needed in this case, and it came from questioning accepted truths beyond AI training, at least as of now. What about Danielson's vision a year from now?

The Clay Navier-Stokes problem solved by AI (or not who can tell?) has put mathematicians into free fall and since theoretical physics is mathematics, also the theoretical physicists as witnessed by Danielson. 

Smoothness vs Wellposedness vs Clay Problem vs Ghost Solutions

The official Clay Navier-Stokes Prize Problem asks about existence of smooth solutions but does not mention uniqueness or wellposedness as continuous dependence of the solution on data. This is because in the standard mathematical analysis of differential equations, uniqueness/wellposedness is viewed to be a byproduct of proving smoothness through bounds of derivatives of the solution in terms of data.

Asking for smooth solutions in the Clay Problem thus is viewed to include uniqueness without specific mention. But this opens an ambiguity by allowing the continuous dependence to include Lipschtiz or continuity constants of any size. This opens to viewing a turbulent solution with very large derivatives as a smooth solution with very large Lipschitz constants effectively eroding the very meaning of continuous dependence.

The true nature of turbulent flow is non-smooth with continuous dependence of mean values but not point values, thus exhibiting a form of weak wellposedness allowing meanvalues such as drag and lift to be computed as well determined quantities with continuous dependence on data. 

The official formulation by Fefferman as leading mathematical analyst thus reflects a form of mathematical analysis, which misses the main character of the Navier-Stokes equations with small vanishing viscosity as having turbulent solutions, which are non-smooth without singularities, thus falling outside the Fefferman formulation. 

Fefferman's mistake was to set viscosity to unity for a problem with small/vanishing viscosity as essence. In a math test that would give an F enforced by the fact that the Euler equation with zero viscosity is mentioned in the same breath as Navier-Stokes. The Clay problem debacle shows the effect of separating math from physics in a problem with physics origin. 

The Clay problem thus needs a reformulation bringing in the turbulence very clearly expressed as motivation, but then forgotten. Without reformulation the problem will continue to direct major efforts into capturing solutions of no interest, with the recent OpenAI solution now shown by Constantin et al to be a ghost solution.    

 

Clay Math Institute Response to Criticism of Navier Stokes Problem Formulation

Response from Martin Briden President Clay Mathematics Institute:

Dear Prof Johnson

I am familiar with the objections that you raise concerning the mathematical problem posed as the Millennium Prize Problem related to Navier-Stokes. Other applied mathematicians and engineers have expressed similar opinions. But I do not accept your conclusion that the problem “cannot be given a meaningful solution”.

I don’t think it has ever been claimed that the problem as posed dealt with turbulence in physical fluids, as you want to see addressed. It is nevertheless surely a natural and compelling question in the study of PDE. 

It may not be the problem that you would have wished to see as a Prize Problem, but from my point of view it has served its purpose well. 

Kind regards

Martin Bridson

My response: 

Dear Martin

Thank you for quick response, and acknowledgment that you are familiar with my criticism of the official formulation of the Navier-Stokes problem, which with the OpenAI solution gets new actuality. 

You say that the problem formulation has served Clay Institute well. I do not think this is so with now the AI solution presenting a potentially disastrous ground shot to mathematics as we know it: An AI proof to a problem without real meaning from physics and mathematics point of view, a 167 page proof which no mathematician can inspect in detail and so will have to be evaluated by the AI that has produced it. 

No mathematician has made any comment as if nothing has happened. But what will now be the fate of the Clay NS problem? Declared as solved but without any meaning? Debunked as nonsense AI with the problem remaining unsolved until next round of AI proof with doubled page number? How do you plan to handle the situation? Have you consulted with the experts behind the problem formulation? What do they say? How will the Clay Math Institute handle the new world of AI mathematics?

Sincerely

Claes

Letter to Clay Math Institute on Navier-Stokes Problem

I have today sent the  following letter to Clay Mathematics Institute including this analysis of the OpenAI solution.  Copies to Terence Tao, Peter Constantin and Charles Fefferman (formulated the problem).

President 

Clay Mathematics Institute

This is a follow up of my 2014 letter about the formulation of the Navier-Stokes problem motivated by the proposed OpenAI solution, which is expressed in the enclosed manuscript. I hope you will read and return with a comment after consulting with the experts. 

In short, my analysis exhibits the unfortunate elimination of turbulent fluid flow, the essence of Navier-Stokes from both physical and mathematical point of view, from the official formulation of the problem, which now has triggered an AI solution without both physical and mathematical meaning. Clay Mathematics Institute is not served by announcing prize problem missing the essentials.

Sincerely

Claes Johnson 

On Fri, 16 May 2014 at 16:50, Claes Johnson <claesjohnson@gmail.com> wrote:

President

Clay Mathematics Institute

I want to convey the information that the formulation of the Clay Navier-Stokes problem is incorrect both mathematically and physically, because the fundamental aspects of (i) wellposedness and (ii) turbulence, are not included, as exposed in detail in the following sequence of blog posts:

http://claesjohnson.blogspot.se/search/label/clay%20problem

The result is that the problem cannot be given a meaningful solution and thus does not serve well as a Prize problem. Evidence is given by the fact that no progress towards a solution has been made.

I have tried to engage Charles Fefferman, who has formulated the problem, Peter Constantin, who acts as a referee, and Terence Tao, who is working on the problem, into a discussion, but I get no response.

I hope this way to stimulate discussion, which I think would be more constructive than no discussion.

Sincerely

Claes Johnson

prof of applied mathematics 

Royal Institute of Technology, Stockholm